Can neural networks be used to prove conjectures?
Your idea may be feasible in general, but a neural network is probably the wrong high level tool to use to explore this problem.

A neural network's strength is in finding internal representations that allow for a highly nonlinear solution when mapping inputs to outputs. When we train a neural network, those mappings are learned statistically through repetition of examples. This tends to produce models that interpolate well when given data similar to training set, but that extrapolate badly. 

Neural network models also lack context, such that if you used a generative model (e.g. an RNN trained on sequences that create valid or interesting proof) then it can easily produce statistically pleasing but meaningless rubbish. 

What you will need is some organising principle that allows you to explore and confirm proofs in a combinatorial fashion. In fact something like your idea has already been done more than once, but I am not able to find a reference currently.

None of this stops you using a neural network within an AI that searches for proofs. There may be places within a maths AI where you need a good heuristic to guide searches for instance - e.g. in context X is sub-proof Y likely to be interesting or relevant. Assessing a likelihood score is something that a neural network can do as part of a broader AI scheme. That's similar to how neural networks are combined with reinforcement learning.

It may be possible to build your idea entirety out of neural networks in principle. After all, there are good reasons to suspect human reasoning works similarly using biological neurons (not proven that artificial ones can match this either way). However, the architecture of such a system is beyond any modern NN design or training setup. It definitely will not be a matter of just adding enough layers then feeding in data.